Quantum Selection of Order in an $XXZ$ Antiferromagnet on a Kagom\'e Lattice

Selection of the ground state of the kagom'e-lattice $XXZ$ antiferromagnet by quantum fluctuations is investigated by combining non-linear spin-wave and real-space perturbation theories. The two methods unanimously favor $\mathbf{q}=0$ over $\sqrt{3}\times \sqrt{3}$ magnetic order in a wide range of the anisotropy parameter $0\le \mathrm{\Delta }\le 0.72$. Both approaches are also in an accord on the magnitude of the quantum order-by-disorder effect generated by topologically non-trivial, loop-like spin-flip processes. A tentative $S-\mathrm{\Delta }$ phase diagram of the model is proposed.